One significant difference you will notice is the reduced number of round-trips when performing a full handshake. Older versions of the TLS protocol require two complete round-trips before the client sends the application data. With TLS v1.3 only 1 round-trip is required! Additionally, the server can send application data in response to the clients first handshake message! This means network latency has less impact on the time required to establish a secure connection.
Another difference is the way session resumption works in TLS v1.3. Previous versions of TLS have the client send a session id which the server has to lookup in its cache. If there was a match then they used the same security parameters. This is a very simplistic mechanism that requires sharing of state on servers.
TLS v1.3 has made significant improvements by re-purposing the ticketing system tacked onto older versions of TLS. The server sends the client a new session ticket after the handshake is complete. This ticket, a blob of data to the client, can be a database lookup key like the old session id. Alternatively, it can be a self-encrypted and self-authenticated value that contains the data for the previous connection. This means the server can be stateless!
Finally, the specification has been evaluated by cryptographic experts in efforts to prove the security of the protocol. While no security proof is perfect, the previous attacks on renegotiation, protocol version downgrading, compression, CBC and padding have been mitigated and the protocol is generally more resistant to attack.
For more details on using TLS v1.3 with wolfSSL, please contact us at email@example.com